{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:KU2C52MOAXFM2SCEBGKZNPZZJZ","short_pith_number":"pith:KU2C52MO","canonical_record":{"source":{"id":"1601.03342","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-01-13T18:36:08Z","cross_cats_sorted":[],"title_canon_sha256":"900b70fa24c77923b71c15bf39e766f188f65debc2270f4954159777fd5e5321","abstract_canon_sha256":"a9fd616fbf78c060b7eb91ff55078abc6da042766810bcde22e66bf82c563565"},"schema_version":"1.0"},"canonical_sha256":"55342ee98e05cacd4844099596bf394e417957295a4b4de1071fca7c18bb74cf","source":{"kind":"arxiv","id":"1601.03342","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.03342","created_at":"2026-05-18T01:22:54Z"},{"alias_kind":"arxiv_version","alias_value":"1601.03342v1","created_at":"2026-05-18T01:22:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03342","created_at":"2026-05-18T01:22:54Z"},{"alias_kind":"pith_short_12","alias_value":"KU2C52MOAXFM","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"KU2C52MOAXFM2SCE","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"KU2C52MO","created_at":"2026-05-18T12:30:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:KU2C52MOAXFM2SCEBGKZNPZZJZ","target":"record","payload":{"canonical_record":{"source":{"id":"1601.03342","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-01-13T18:36:08Z","cross_cats_sorted":[],"title_canon_sha256":"900b70fa24c77923b71c15bf39e766f188f65debc2270f4954159777fd5e5321","abstract_canon_sha256":"a9fd616fbf78c060b7eb91ff55078abc6da042766810bcde22e66bf82c563565"},"schema_version":"1.0"},"canonical_sha256":"55342ee98e05cacd4844099596bf394e417957295a4b4de1071fca7c18bb74cf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:54.738950Z","signature_b64":"lhKP9JsM8EdvwqfguxTeICmQu0OzjUXxXpdX5KYj5he8GkTicSQUvGHwOXSmRkPaQrqWG1pRdLyGBJLdFK+cDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"55342ee98e05cacd4844099596bf394e417957295a4b4de1071fca7c18bb74cf","last_reissued_at":"2026-05-18T01:22:54.738419Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:54.738419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.03342","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:22:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kx1g4rKmFEan6ZPjHcq0q9+Jg7vhC9LOchh6CK2tKfxbZ6nVPo2Afdz9toh7eIamAp1tuMbKxh15KMaXEFefDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T23:15:18.806970Z"},"content_sha256":"9db54db914d5edcaa7f5a4cdd914f38ddc6591735690fe46294f33bedc4761ad","schema_version":"1.0","event_id":"sha256:9db54db914d5edcaa7f5a4cdd914f38ddc6591735690fe46294f33bedc4761ad"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:KU2C52MOAXFM2SCEBGKZNPZZJZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting Mapping Class group orbits on hyperbolic surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Maryam Mirzakhani","submitted_at":"2016-01-13T18:36:08Z","abstract_excerpt":"Let $S_{g,n}$ be a surface of genus $g $ with $n$ marked points. Let $X$ be a complete hyperbolic metric on $S_{g,n}$ with $n$ cusps. Every isotopy class $[\\gamma]$ of a closed curve $\\gamma \\in \\pi_{1}(S_{g,n})$ contains a unique closed geodesic on $X$.\n  Let $\\ell_{\\gamma}(X)$ denote the hyperbolic length of the geodesic representative of $\\gamma$ on $X$. In this paper, we study the asymptotic growth of the lengths of closed curves of a fixed topological type on $S_{g,n}.$ As an application, one can obtain the asymptotics of the growth of $s^{k}_{X}(L)$, the number of closed curves of length"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03342","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:22:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OnucsZqHApHcDGSDuQ39s8ZdG3Z85elTw6qHWKxIkt3N4yqiNFsqAO/6C0OUGne8mdywWn0H25/hfi/eS/bpDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T23:15:18.807368Z"},"content_sha256":"a8deaa29bb318b3bfd75036cef138a107bb4cdef5c16ffe471e3f3f0becd79e4","schema_version":"1.0","event_id":"sha256:a8deaa29bb318b3bfd75036cef138a107bb4cdef5c16ffe471e3f3f0becd79e4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KU2C52MOAXFM2SCEBGKZNPZZJZ/bundle.json","state_url":"https://pith.science/pith/KU2C52MOAXFM2SCEBGKZNPZZJZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KU2C52MOAXFM2SCEBGKZNPZZJZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-26T23:15:18Z","links":{"resolver":"https://pith.science/pith/KU2C52MOAXFM2SCEBGKZNPZZJZ","bundle":"https://pith.science/pith/KU2C52MOAXFM2SCEBGKZNPZZJZ/bundle.json","state":"https://pith.science/pith/KU2C52MOAXFM2SCEBGKZNPZZJZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KU2C52MOAXFM2SCEBGKZNPZZJZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:KU2C52MOAXFM2SCEBGKZNPZZJZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"a9fd616fbf78c060b7eb91ff55078abc6da042766810bcde22e66bf82c563565","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-01-13T18:36:08Z","title_canon_sha256":"900b70fa24c77923b71c15bf39e766f188f65debc2270f4954159777fd5e5321"},"schema_version":"1.0","source":{"id":"1601.03342","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.03342","created_at":"2026-05-18T01:22:54Z"},{"alias_kind":"arxiv_version","alias_value":"1601.03342v1","created_at":"2026-05-18T01:22:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03342","created_at":"2026-05-18T01:22:54Z"},{"alias_kind":"pith_short_12","alias_value":"KU2C52MOAXFM","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"KU2C52MOAXFM2SCE","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"KU2C52MO","created_at":"2026-05-18T12:30:29Z"}],"graph_snapshots":[{"event_id":"sha256:a8deaa29bb318b3bfd75036cef138a107bb4cdef5c16ffe471e3f3f0becd79e4","target":"graph","created_at":"2026-05-18T01:22:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $S_{g,n}$ be a surface of genus $g $ with $n$ marked points. Let $X$ be a complete hyperbolic metric on $S_{g,n}$ with $n$ cusps. Every isotopy class $[\\gamma]$ of a closed curve $\\gamma \\in \\pi_{1}(S_{g,n})$ contains a unique closed geodesic on $X$.\n  Let $\\ell_{\\gamma}(X)$ denote the hyperbolic length of the geodesic representative of $\\gamma$ on $X$. In this paper, we study the asymptotic growth of the lengths of closed curves of a fixed topological type on $S_{g,n}.$ As an application, one can obtain the asymptotics of the growth of $s^{k}_{X}(L)$, the number of closed curves of length","authors_text":"Maryam Mirzakhani","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-01-13T18:36:08Z","title":"Counting Mapping Class group orbits on hyperbolic surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03342","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:9db54db914d5edcaa7f5a4cdd914f38ddc6591735690fe46294f33bedc4761ad","target":"record","created_at":"2026-05-18T01:22:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a9fd616fbf78c060b7eb91ff55078abc6da042766810bcde22e66bf82c563565","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-01-13T18:36:08Z","title_canon_sha256":"900b70fa24c77923b71c15bf39e766f188f65debc2270f4954159777fd5e5321"},"schema_version":"1.0","source":{"id":"1601.03342","kind":"arxiv","version":1}},"canonical_sha256":"55342ee98e05cacd4844099596bf394e417957295a4b4de1071fca7c18bb74cf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"55342ee98e05cacd4844099596bf394e417957295a4b4de1071fca7c18bb74cf","first_computed_at":"2026-05-18T01:22:54.738419Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:54.738419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lhKP9JsM8EdvwqfguxTeICmQu0OzjUXxXpdX5KYj5he8GkTicSQUvGHwOXSmRkPaQrqWG1pRdLyGBJLdFK+cDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:54.738950Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.03342","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9db54db914d5edcaa7f5a4cdd914f38ddc6591735690fe46294f33bedc4761ad","sha256:a8deaa29bb318b3bfd75036cef138a107bb4cdef5c16ffe471e3f3f0becd79e4"],"state_sha256":"2021393895ae89380013189e77967bbf4818929e42a54bf6e165ad6963a33476"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b9fPhhpV0Exo2n7F4rcHfrOAfuLnqYUoy6YCPPxFrjmdgxU6y5oYZvvQ0LwP7O8XRRoqUdYiiGTAwy5r2YZjDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T23:15:18.810047Z","bundle_sha256":"e22e69f0a1a964667fc9d8e5c77f5bc8f9b1e90d9e4b3342a00fab94e5ffe4db"}}